We introduce a new ideal so called the residue ideal of the logarithmic differential forms that describes the behavior of poles along a hyperplane. By using this ideal, we give a description of the logarithmic differential 1-forms in terms of the cover ideal of graphs, which is an object appearing the Stanley-Reisner theory. By using the Stanley-Reisner theory, we give several new results on invariants of these modules.
This is a joint work with Satoshi Murai.
A plane projective curve of degree d is hyperelliptic is a curve having a point of multiplicity d-2. We study the moduli and realization spaces of these curves using standard hyperelliptic curves in weighted projective planes. These standard curves are part of a special pencil which induce pencils containing the hyperellicptic curve and the necessary information to study the fundamental group of the complement. We detect when these groups are non-abelian and we study when the fundamental groups are finite, with an almost-complete answer for the following question: which degrees admit a plane algebraic curve with finite non-abelian fundamental group.
The coordinate ring of the Grassmannian Gr(k,n) of k dimensional subspaces of C^n has the structure of a cluster algebra. An additive categorification of it has been given by Jensen-King Su, using a quotient of a preprojective algebra of a cyclic quiver. This algebra is infinite dimensional and its module categories is wild in general. We show how this cluster category can be given using dimer models in the disk (certain quivers with faces) and describe the cluster categories using combinatorial tools.
We consider the following question: Does the family of all smooth hyperplane sections of a smooth irreducible projective hypersurface vary maximally in moduli? As observed by Beauville in 2025, the question is equivalent to the Weak Lefschetz Property of the Jacobian ideal in the hypersurface degree. We translate the problem into the vanishing of the cohomology of a suitable stable syzygy bundle. By a joint work with R. M. Mirò-Roig, we can give a positive answer in sufficiently high degree.
Very recently the result has been extended to certain classes of projective varieties by Favale and Pirola, and to hypersurfaces of lower degree with possibly isolated singularities by Ilardi, Nasrollah Nejad, and Tafazolian.
This talk is part of an ongoing joint work with Eva Elduque (UAM).
Our purpose is to connect and unify two types of results regarding the structure of quasi-projective groups, that is, fundamental groups of quasi-projective manifolds. Based on the classical Castelnuovo-De Franchis theorem for K\"ahler manifolds, the following well-known problem is considered, which will be referred to as the Geometric Morphism Problem.
Geometric Morphism Problem:
Let $U$ be a quasi-projective manifold and \psi : \pi_1(U)\to G$ be an epimorphism to a smooth complex curve group $G$ with a finitely generated kernel. Determine when/if there exists a morphism $F : U \to C$ to a smooth complex curve $C$ realizing $\psi$, that is, such that $\psi= F_∗ : \pi_1 (U) \to \pi_1(C) = G$, up to isomorphism in $\pi_1(C)$.
We will state an extension of this problem using orbifold structures. In particular, denote by $\mathbb{G}_{g,(r,\bar{m})}$ the orbifold fundamental group of a Riemann surface of genus $g$ with $r$ punctures and $s$ orbifold points with orders $\bar{m} = (m_1,\dots,m_s)$. One can state the following analogue of the Geometric Morphism Problem.
Geometric Orbifold Morphism Problem:
Let $U$ be a quasi-projective manifold and $\psi:\pi_1(U) \to \mathbb{G}_{g,(r,\bar{m})}$ be an epimorphism with finitely generated kernel. Determine when/if there exists an orbifold morphism $F : U \to C_\varphi$ to a smooth complex curve $C$ with orbifold structure $C_\varphi$ realizing $\psi$, that is, such that $\psi= F_∗ : \pi_1 \to \pi_1^{\textrm{orb}}(C_\varphi) = \mathbb{G}_{g,(r,\bar{m})}$, up to isomorphism in $\pi_1^{\textrm{orb}}(C_\varphi)$.
We will discuss an answer to the Geometric Orbifold Morphism Problem in terms of the following invariant of the group $\chi_{g,(r,\bar{m})} := 2 - 2g - r - \sum_{i=1}^s \left( 1-\frac{1}{m_i}\right)$.
Theorem:
The Geometric Orbifold Morphism Problem has a positive answer for $\mathbb{G}_{g,(r,\bar{m})}$ if $\chi_{g,(r,\bar{m})} < 0$. Moreover, the multiple fibers of $F$ are determined by $\bar{m}$ and one such $F$ is uniquely determined up to isomorphism of algebraic varieties in the target.
We will also discuss how sharp the Theorem is in the case $\chi_{g,(r,\bar{m})} \geq 0$ and present some applications.
In this talk, I will discuss some recent advances in the study of moments of L-functions. This is based on joint work with Bergström--Petersen--Westerland, and Miller--Patzt--Petersen--Wang.
In this talk I will present two parallel constructions of asymptotic expansions in Hitchin theory and the ones in Habiro rings. It is based on joint work on quantum curves with Motohico Mulase and the geometry of the asymptotic expansions of Nahm sums in Habiro rings of Garoufalidis, Scholze, Wheeler and Zagier. The talk is also inspired by the discussions and work with Angela East and Kexuan Yang.
We consider some compactifications of the moduli spaces of curves and study their fundamental groups by means of quantum representations, thereby providing new examples of Kahler groups. We further derive constraints for the images of fundamental groups of curves under complex algebraic maps to the moduli of curves. (joint work with Philippe Eyssidieux)
The property (T) conjecture for mapping class groups predicts that finite-dimensional unitary local systems on the moduli spaces of curves $\mathcal{M}_{g,n}$ for $g\geq 3$ are rigid (in the sense that they admit no infinitesimal deformations). While this question is fairly well studied for local systems with finite monodromy, a special case known as Ivanov's conjecture, little is known when the monodromy is infinite.
We establish the rigidity of local systems of conformal blocks arising from the $SU(2)$ and $SO(3)$ modular categories, over $\mathcal{M}_g$ for $g\geq 7$ and at conformal levels $\ell$ such that $\ell+2$ is prime and greater than or equal to $5$. These are natural examples with infinite monodromy arising in quantum topology via the Witten-Reshetikhin-Turaev construction, or alternatively in conformal field theory via the Wess-Zumino-Witten model.
The heart of the argument is a proof that any infinitesimal deformation of a local system of conformal blocks, within the space of all flat unitary local systems, necessarily remains a local system of conformal blocks. This then implies triviality, since local systems of conformal blocks admit no internal deformation of this kind, by a result called Ocneanu rigidity. The proof combines the factorization property of conformal blocks with elementary Hodge theory on certain root stacks over $\overline{\mathcal{M}}_{g,n}$, over which the local systems of conformal blocks extend.
I will explain how a “q-deformed” version of Borel–Laplace resummation allows us to interpolate the coloured Jones polynomial to non-integer colours. I will describe how the method works in the case of the figure-eight knot. This is an ongoing joint project with Vladimir Mangazeev and Marcos Mariño.
For $\Sigma_{g,n}$ a genus $g$ surface with $n$ punctures, there is an action of the mapping class group on the set of representations of $\pi_1(\Sigma_{g,n})$. In joint work with Josh Lam and Daniel Litt, we aim to classify the representations with finite orbit under this mapping class group action, which we call canonical representations.
Hurwitz spaces are important moduli spaces that parameterize maps between algebraic curves, and play an important role in algebraic geometry and number theory. They also arise naturally in topology as certain covers of configuration spaces of points on a surface. I will discuss joint work with Aaron Landesman on understanding homological stability for a very general class of Hurwitz spaces, extending the work of Ellenberg--Venkatesh--Westerland, who had proven it under restricted hypotheses.
For V_g a 3-dimensional genus g handlebody, the genus g handlebody group is the group of isotopy classes of self-diffeomorphisms of V_g. Two fundamental representations of this group are the first homology groups of V_g and of its boundary surface. In this talk, I will discuss joint work in progress with Arthur Soulié, where we calculate the cohomology groups of the handlebody group with coefficients in tensor products of these representations and their dual representations, in the stable range (i.e. for g sufficiently large compared to the cohomological degree).
To any discrete group G, one can associate a graded Lie algebra Lie(G) by considering the associated graded of the lower central series of G. We will begin by recalling this classical construction, rooted in combinatorial group theory, before focusing on the case of the Torelli group I(S) of a compact oriented surface S.
Then, we will review the foundational work of D. Johnson on the abelianization of I(S), as well as the result of R. Hain on the rational structure of Lie(I(S)). Next, using elementary methods, we will show that, unlike the degree 1 case, the degree 2 part of Lie(I(S)) is torsion-free. Finally, we will explain how, following and refining recent results by Y. Nosaki, M. Sato, and M. Suzuki, torsion can be explictly identified in higher degrees using the LMO functor (a "universal" invariant of 3-dimensional cobordisms arising from quantum topology). This talk is based on past and new works in collaboration with Q. Faes and M. Sato.
We use Gay and Kirby’s description of 4-manifolds in terms of trisections and trisection diagrams to define a 4-manifold invariant.
The algebraic data are an indecomposable finite semisimple bimodule category with a bimodule trace over a pair of spherical fusion categories and a pivotal functor from another spherical fusion category into the spherical fusion category of its bimodule endofunctors and natural transformations between them.
The resulting 4-manifold invariant is formulated in terms of diagrammatic calculi and includes the earlier invariants of Bärenz and Barrett and of Chaidez, Cotler and Cui as special cases.
This is joint work with Vincentas Mulevičius and Fiona Torzewska, arXiv:2511.19384
The non-Abelian Hodge correspondence (NAH, Simpson et al.) is a categorical equivalence between the Dolbeault moduli space of Higgs bundles and the de Rham moduli space of holomorphic connections, both defined on a smooth complex projective curve and with values in the same complex simple Lie group, which underlies a homeomorphism between the moduli spaces. This is a transcendental map that cannot be concretely evaluated. Gaiotto proposed a scaling limit of NAH from his physics considerations. In a previous paper with Dumitrescu, Fredrickson, Kydonakis, Mazzeo and Neitzke, we proved that Gaiotto's limit exists, and gives a concretely calculable map to produce opers. The purpose of this talk is to show that the inverse of Gaiotto's map is precisely the map from the moduli of $^LG$-opers to the $G$-Hitchin base that is used in the Geometric Langlands Correspondence for the oper case (Beilinson-Drinfeld). The concrete evaluations and comparison of the maps are possible due to the SL(2) behind the scenes, and Kostant's TDS appearing in both stories.
(This research was supported by NSF-FRG and MPI-MiS in Leipzig.)
It is well known that the complements of many knots and links possess a hyperbolic structure. A standard approach to investigating a complement is to decompose it into ideal tetrahedra. In this talk, we present a new method for decomposing the complement using complexified hyperbolic tetrahedra instead of ideal ones. This decomposition is constructed from a knot diagram and an SL(2, C) representation of the knot group. Each complexified tetrahedron corresponds to a quantum 6j-symbol arising from the sl(2) quantum group, providing a new approach to proving the volume conjecture, which relates the hyperbolic volume of the complement to the colored Jones polynomial.
The Milnor-Wood inequality, introduced in two landmark papers by John Milnor(1958) and John W. Wood (1971), is a striking result at the intersection of geometry, topology, and dynamics. It establishes sharp bounds on the Euler number of flat S^1-bundles over surfaces, revealing deep connections between geometric curvature and topological invariants. Milnor’s original inequality highlights the boundedness of Euler invariants for flat bundles with "linear" structures, which Gromov later generalized using bounded cohomology.
Wood extended Milnor's result to "non-linear" flat circle bundles, offering a perspective rooted in 1-dimensional dynamics. In the 1980s, Étienne Ghys posed the intriguing question of whether Wood’s inequality could be extended to flat-oriented S^3-bundles. In this talk, we will also discuss the surprising ways in which the inequality fails in higher-dimensional non-linear cases, showcasing the new calculations in the bounded cohomology of diffeomorphism groups.
We associate a cluster algebra to every knot or link in such a way that the Alexander polynomial of the knot is recovered from certain cluster variables under specialization. More precisely, given a link diagram K, we define a quiver Q whose vertices correspond to the segments of K and whose arrows go clockwise around the crossing points of K. For every segment i of K, we construct an indecomposable quiver representation T(i) by defining its vector spaces and linear maps. In the cluster algebra, we construct corresponding cluster variables x(i) by specifying an explicit mutation sequence. In particular, the collection of all x(i) forms a cluster in the cluster algebra. Each x(i) (and the F-polynomial of each T(i)) specialize to the Alexander polynomial of K, under the same specialization.
The truncations of the surface group ring by powers of its augmentation ideal form representations of the mapping class group which have been shown to often build up entirely the representations arising from configuration spaces on the surface. To what extent does this picture generalise to other representations of the mapping class group? We will survey recent results on this question in the context of configuration spaces, and then focus on the truncated group rings themselves, with particular emphasis on their relation to the twisted cohomology of the mapping class group.
Moduli spaces of surfaces somewhat surprisingly give rise to operads that detect infinite loop spaces and that can be used to construct infinite loop maps. Motivated by these and a construction by Boekstedt, we will present a general construction of infinite loop spaces and maps between them.
This is ongoing joint work with Simon Gritschacher.
Functors on the category of free groups appear in several areas of algebra and topology. Understanding Ext-groups between such functors is a natural problem, motivated in part by the study of the stable homology of automorphism groups of free groups.
For the tensor powers of the abelianization functor, integral Ext-groups can be computed using an explicit resolution of the abelianization functor together with a Künneth formula. From these computations one obtains, in particular, that the rational Ext-groups between the abelianization functor and its symmetric powers vanish.
Recently, Arone used homotopical methods to compute integral Ext-groups between functors of free groups. He constructed a complex whose homology gives the integral Ext-groups between the abelianization functor and its symmetric powers. Arones’s results show that these Ext-groups are far more intricate than their rational counterparts.
In this talk, I will revisit Arone’s complex from a functorial perspective and show how it can be used to derive explicit computations. As an application, I will present joint work with Minkyu Kim determining the first and second Ext groups between the abelianization functor and its symmetric powers. I will also describe ongoing joint work with Gregory Arone and Minkyu Kim establishing a surprising connection between integral Ext groups between the abelianization functor and its symmetric powers and the homology of braid groups.
George Altmann (University of Leeds) — A peripheral structure for welded links from the free loop space of their ribbon torus complements
Ari Davidovsky (University of Wisconsin-Madison) — The Homology of the Level 4 Braid Group
Nicolas Guès (Université Sorbonne Paris Nord) — Tame homotopy type of knot spaces
Vladimir Ivanović (University of Montenegro) — Z2-homology of the orbit spaces Grs(n)/T^n and the moduli space of weighted pointed stable curves
Butian Zhang (Université de Toulouse) — 1-Cohomology Classes from the Long Knot Space to the Closed Knot Spaces